From the book: General principle: We can define a property of any smooth surface provided we can define it for any surface patch in such a way that it is unchanged when the patch is reparametrized. That so doesn't sound right. Say I have a smooth surface S and an atlas of regular surface patches. I then define a property X of a surface patch and verify that this property is independant of a change of parametrization. According to the principle, I have unambiguously defined property X for the smooth surface S itself. But say for exemple that the atlas of S is made of two surface patches #1 and #2 that map to distinct areas of S. Now suppose that according to our definition of X, surface patch #1 is X while surface patch #2 is not X. What do we say about S, is it X or not? I must be misinterpreting the "general statement". What does it mean to you?